On Friday, 24 June 2016 at 03:22:11 UTC, Timon Gehr wrote:
On 24.06.2016 04:36, Smoke Adams wrote:
....
You do realize that e^(-1/t)^t is a counter example?
e^(-1/t) -> 0 as t -> 0
t -> 0 as t -> 0
....
That's not a counterexample to anything I said. ^ is
discontinuous at (0,0) and indeed, you can force the limit to
an arbitrary value by choosing the two expressions the right
way. That's clear.
but e^(-1/t)^t does not -> 1 as t-> 0, which is obvious since
it/s 1/e.
So, We can define 0^0 = 1 and maybe that is better than 2, but
it is
arbitrary in the sense that it's a definition. It may bear
fruit but it
is not necessarily meaningful.
...
It's meaningful in those cases where you want to use 0^0, and
otherwise just don't use it.
Suppose a person is running some numerical simulation that
happens to be
computing an a value for an equation that is essentially based
on the
above.
Then the numerical simulation is inherently broken. a^b takes
on any arbitrary non-negative value in an arbitrary small
region around (0,0) and is undefined at many points in such a
region. (BTW: It would be fine with me if 0.0^^0.0 was NaN --
that's a completely different case than the one at hand: pow on
integers.)
... break the laws of physics by
arbitrarily defining something to be true when it is not.
...
Utter nonsense. (Also note that the 'laws of physics' typically
give rise to piecewise analytic functions, and if you only
consider analytic functions, 0 ^ 0 = 1 is actually the right
answer.)
Please, it seems you only know enough about math and physics to
get you into trouble.
1. do you realize that the "equations" of man are only
approximations to the equations of life? Or do you really think
gravity behaves exactly as 1/r^2?
Also, what about when r = 0? how do you propose to define it? Set
it to 0? 1? -infinity? What's the solution here? Surely there is
a non-arbitrary solution?
2. You hold way to much faith in man. Math and Science are
tools, tools are created, manipulated, updated, retooled, and
used to make other tools. Their only use is how well they work.
They work quite well but we cannot prove how accurate they are.
If you know about Godel, you would know that we can't even prove
that we can't prove anything(which itself is meaningless).
3. What you are arguing is the convenience factor. Sometimes also
known as the lazy factor. Just admit it and stop trying to prove
that somehow god has made this choice.
4. An equation in math is just one form of the same abstract
mathematical construct. Simply put: f(x) = x = sqrt(x)^2 =
inv(cos(x)) = .... All these types of identities can substituted
in an equation to change it's form.
5. All forms are different. Obviously x != sqrt(x)^2 in all
cases. Hence the transformations using identities are not
actually equal in all cases: sqrt(x)^2 = x only for x >= 0.
6. Symbols are not absolute. They are mans creation and agreement
on their meaning so they can be useful. x is just chicken scratch
until everyone agrees on how to interpret it.
7. And this is where the problem lies. Assumptions can be deadly.
They should only be made when they have to. Your argument about
using integers only is nonsense because you have made the
assumption that only integers will ever be used and also that the
integers are not related to the real number case. Both of these
are provably false.
e.g., Lets suppose that, for integers, we have PowI(x,y) and for
reals, PowR(x,y).
Someone does
if (isInteger(x) && isInteger(y)
return PowI(x,y);
else
return PowR(x,y);
Why they might do this is immaterial.. I just did it, so it is
possible. Maybe PowI is faster because of some type of new
algorithm someone came up with.
8. The truth is the correct answer. The truth of the matter is,
0^0 is undefined. Accept it, just because you don't like the
answer doesn't give you the right to change it. This is the same
belief many have in Santa Clause, the Tooth Fairy, Jesus Christ,
etc. No proof they exist but they belief it and change the truth
to match the result(which is anti-science, anti-math, and
anti-logic, anti-progress, and anti-truth and ultimately
anti-god).
It's one thing to argue the convenience factor, and this is ok
but it should be well understood by everyone. The other is to
argue that what you are saying is fact, and this is clearly
demonstrably false. Arguing a special case, such as using only
integers, is just as false because F => F is a false statement.
9. If you are using some other kind of logical system than the
standard one that all of math and science is built on, then
forgive me for being wrong... But you didn't state this from the
beginning and I mistakenly assumed we were in the same reality.
